Stochastic methods for physics and biology, spring 2010
Lecturer
Stefan Geritz
Paolo MuratoreGinanneschi
Scope
The aim of the course is to introduce the basic concepts of the theory of stochastic differential equations needed in applications (applied mathematics, physics and biology). In particular we will illustrate methods of qualitative, asymptotic and numerical analysis.
10 cu.
Type
Advanced studies.
Prerequisites
Lectures
Weeks 39 and 1118, Monday 1416 in room C124 and Friday 1416 in room C123.
Easter holiday 1.7.4.
First lecture: Friday, 22.01.2010
Lecture Notes
The lecture notes cover and sometimes integrate the material expounded in the lections. They also give bibliographic references for the same topics. For exercises please refer to the section below.
Lectures 110  Lectures 1020 

Lecture_01: Background of Probability  Lecture_11: KolmogorovChentsov theorem 
Lecture_02 (rev 29.01): Bernoulli variables, BorelCantelli lemma  Lecture_12: Whitenoise 
Lecture_03: Limit theorems  Lecture_13: Ito integral 
Lecture_04: Martingales  Lecture_14: Ito and Stratonovich calculus 
Lecture_05: Numerical generation of random variables  Lecture_15: SDE, existence and uniqueness 
Lecture_06 (rev 12.02): Statistical tests  Lecture_16: Kolmogorov pair and diffusion 
Lecture_07: FokkerPlanck equation I  Lecture_17: Exit time statistics 
Lecture_08: FokkerPlanck equation II  Lecture_18: Girsanov formula 
Lecture_09: Brownian motion  Lecture_19: Numerical integration of SDE's 
Lecture_10: KarhunenLoève representation  Lecture_20: Population dynamics 
Probabilistic interpretation of set theoretic concepts
Exams
There will be two options for taking the course. Passing an exam to be held after the end of the course, in
date to be agreed or working out a short project (recommended).
Bibliography
 L.C. Evans, "An Introduction to Stochastic Differential Equations", lecture notes.
 P. E. Kloeden, E. Platen, H. Schurz, "Numerical Solution Of Sde Through Computer Experiments", Springer (Universitext) (2003).
 N. Berglund, B. Gentz "NoiseInduced Phenomena in SlowFast Dynamical Systems. A SamplePaths Approach", Springer (Probability and its Applications) (2005).
 D.J. Higham, "An algorithmic introduction to numerical simulation of stochastic differential equations" SIAM Review, Education Section, 43, 2001, 525546. (Link to Higham's publications page.)
Book extended preview is available on line from google books
Registration
Did you forget to register? What to do.
Exercise groups
Group  Day  Time  Place  Instructor 

1.  Wednesday  1618  C124 

Exercise set 1
Exercise set 2
Exercise set 3
Exercise set 4
Exercise set 5
Courseware
The codes below make use of the pgplot graphic library and of the GNU GSL
scientific library. You need to install these packages in order to use the codes as they are.